When is cubic texture interpolation helpful?
Let's look at what the two mean and create examples. Our example image will be this (enlarged) $2\times2$ pixel grid:



Linear: In spaces between other pixels, there is a linear gradient of the two mixing colors:



Cubic: In spaces between other pixels, there is a cubic (also called Ease) gradient between the two colors:



When we compare the linear and cubic interpolations, the eased one seems to "bleed" a little bit, creating a much smoother interpolation, but it doesn't leave the original lines as defined, so it will blur the image.

Let's look at it with a more complex image:



Linear:



Cubic:



Essentially, Linear interpolation will hold to the original definition of the shapes in the image, but won't look as pretty. Cubic interpolation will look nicer and smoother, but that comes at the cost of image clarity, since the image will appear blurred.

Also, when I did (very) small test renders on my computer, Linear was milliseconds faster than Cubic.